Elementarzelle

Elementarzelle is the German term for the unit cell of a crystal, the smallest repeating unit that, through translations, generates the entire crystal lattice. It is defined by three translation vectors a, b, and c and the angles between them, alpha, beta, and gamma. The unit cell volume is given by V = |a · (b × c)|. While every crystal possesses a unit cell, the choice is not unique; a primitive cell contains the minimum number of lattice points, whereas conventional cells are chosen to reflect the lattice’s symmetry and may contain more than one lattice point.

The lattice points contained in a unit cell depend on its centering. A primitive (P) cell has one lattice point per cell. A body-centered (I) cell has two, a face-centered (F) cell four, and base-centered variants (A, B, C) have two. In three dimensions, these centering schemes lead to the 14 Bravais lattices, which classify crystal structures by their translational symmetry.

Elementarzellen are central to crystallography, mineralogy, and materials science. They provide the mathematical framework to describe crystal structures, determine lattice parameters (a, b, c, alpha, beta, gamma), and relate different cells through transformation matrices. They also underpin computational modeling, where the unit cell defines the periodic boundary conditions and the finite simulation box used to study physical properties.

In summary, the elementarzelle is the foundational concept for describing periodic order in crystals, encapsulating geometry, symmetry, and translational repetition in a concise, repeatable unit.